So, in spite all of the incredibly fun and apparently non-productive times I have had in Brazil, I also managed to do quite a bit of reading. Those many days I kept in the library or studying, alongside those long hours in the bus allowed me to tackle quite a bit of material.
One of the things that have been in my attention the last few months are complex numbers. I remember helping out my little sister with some complex number multiplication near the end of last school year and that kind of got me on the subject a little. I had done some exploration of the e^ipi = -1 formula a while ago and had learned to work a little with complex logs but that was just really algorithmic and without any understanding so at the end of spring quarter I decided to look into that topic. Starting with “An imaginary tale: The Story of “i” (by Nahin). If you haven’t read that book, I encourage you to go here and buy it right now. For less than 10 dollars you get hours of entertainment and a pretty good understanding of complex numbers in addition to great explanations of some pretty amazing mathematical magic. After tearing through that book and getting a good grab on most of the equations, I found “Hypercomplex Numbers” at my school (University of Washington) library’s bookshelf. After reading about complex numbers, I also became intrigued about the idea or possibility of “supercomplex” or 3d numbers and made some lame attempts to discover them by myself. hahaha Within a few days, I gave up and cracked open the “Hypercomplex Numbers” book and that was like the discovery of a whole new world to me. Like, for Seattle area people, I would put it into these terms. Redmond would be like just the normal 1-d number line, you know, that is what I grew up with. It is very familiar, but unfortunately, not all that exciting. I would put the complex numbers like me getting my license and getting to know Bellevue and the surrounding areas. Much cooler than just Redmond and I was free to do a lot more things. Then the quaternion system was like learning about Seattle.
Anyways enough of the talk, let me attempt to explain this concept for a non-math reader to get.
For those not familiar with the complex number systems, lets start with the natural numbers.
We can think of them as progressing in a line, like below:
The natural number line
It is a line. One dimensional, fairly-straightforward.
Now with the complex numbers, those weird things that you probably had to learn how to multiply some time in high school for some strange reason that was probably never revealed to you, are pretty straightforward too. The only real thing we have to add/define is the property that i = ( -1 )^2. Without really thinking how something like a negative square root would work, you still can figure out most of the properties of complex numbers, using only that one identity.
For those wanting a quick refresher, here is a short one below. If you feel comfortable with this, feel free to skip it.
i is just another number that you can multiply or add, just with a few added rules to the processes. For example, 4i + 9 is not something you can simplify any further. Those two terms, 4i and 9, are the two parts to only one complex number. Lets call that complex number z. z has both a complex part and a real part. Like everyone learns in 7th grade algebra, you can only add and subtract like terms. That is exactly the same in the complex system.
Multiplication is also pretty easy. For example, 7 * i just becomes 7i. 7i * i becomes 7*(i^2) and since i^2 = (-1), the product 7i * i ends up equaling (-7). Multiplying two complex numbers, lets call them y and z, is just done with the same FOIL (first-outside-inside-last) algorithm that you use with expressions like (4+x)*(5x-11). The only difference is you have to substitute any (i)^2 terms for (-1). So, if you haven’t worked with these for a while, a good exercise is to figure out what the product of complex numbers (a+bi) and (c+di) is.
So here is where the two dimensional extension of the number line comes in. Lets add another axis to our number line and call it the imaginary axis. The only difference with the numbering is that instead of using the real numbers we are used to, we use multiples of i. What is cool about that is now we have a visual way of representing, real, imaginary, and complex numbers on the same graph.
Three numbers plotted on the complex plane
Now that we have already gone over multiplication of two complex numbers, lets think of them as vectors. Since a complex number has two parts (a + bi), we can plot it on the plane. The difference with a vector is that it is not just a point, but rather a line that has a direction to it (represented by an arrow at one end). So lets multiply two numbers and plot them on the complex-plane. We shall do (2+i)*(1+i). The answer is (1+3i).
The multiplication of two complex numbers
Rather than just tell all the rules and properties, I encourage you to figure them out by yourself. Like see what happens if you multiply a complex number by itself. Is there something geometrical you notice that could make the process simpler than going through the FOILing? One thing to note is that “i” can also be referred to as the rotation operator. Why? Because multiplying by “i” rotates a point 90 degrees in the complex plane, enabling you to get back to where you started, just by successive multiplication. So it could be worth it to think about some of the implications of this for square roots, cube roots, and raising something to a power.
The most important part about the addition of “i” though, in my opinion, is the extension of the number system into two dimensions. You can get some really interesting functions with this new 2d system of numbers. It is worth trying to plot something simple (like z+2-i) at every point to see what it does. Is there anything you foresee being difficult about doing this. What dimension graphs do you need to represent a function of the complex numbers?
As you experiment and get a feel for the complex number system, you will see that it makes many two dimensional operations fairly easy. Things like rotations and description of transformations are easy to figure out. Since we live in a 3d world, there was an immediate desire to come up with a 3d system of numbers following the introduction to the complex plane. There is a famous story about the discovery of these and I encourage any reader of this to check out the Wikipedia entries on the Hamiltonian and the Quaternions. See, in order to work in two dimensions, mathematicians only needed to add the imaginary axis to have a complete set of complex numbers. However, things get a little more difficult in 3 dimensions and surprisingly, you end up needing to work with ordered groups of 4 to get an analogous expansion of the number system into 3 dimensions. These are called Quaternions. Well, technically quaternions are made from a four-dimensional basis so they are all 4d (a+bi+cj+dk) but they work really good for 3d problems too.
I am going to talk about Quaternions more tomorrow along with their relative, the octonions! For now, I am going to have a little sleeping break.